On the Eigenstructure of Covariance Matrices with Divergent Spikes

TL;DR

This paper extends the understanding of eigenstructure in covariance matrices with diverging spikes, providing CLTs and eigenvector consistency rates in high-dimensional settings where features and samples grow proportionally.

Contribution

It generalizes previous models by analyzing covariance matrices with diverging spikes, establishing CLTs with flexible centering, and deriving eigenvector consistency rates in high-dimensional regimes.

Findings

Established CLTs for separated diverging spikes in high dimensions.

Showed eigenvector consistency depends on eigenvalue growth.

Filled gaps in the literature for spike numbers growing faster than o(n^{1/6}).

Abstract

For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), \gamma^{-1} \leq \frac{N}{n} \leq \gamma$ where $\gamma \in (0,\infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: \lim_{n \to \infty}{\frac{\sqrt{\log{n}}}{\log{\frac{n}{M(n)}}}}=0.$ Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1/6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.